Abstract
The authors study thermally activated hopping processes on regularly multifurcating ultrametric spaces (UMS) and investigate the first two moments of the range Rn, where n is the number of steps. Paralleling the recent findings for Cayley trees, the knowledge of exact generating functions for these moments permits to draw conclusions about their behaviour for large n. For the first moment, the mean number of distinct sites visited, Sn, they derive analytically the cross-over from a linear behaviour in the transient regime gamma )1 to a sublinear increase Sn approximately ngamma in the recurrent regime gamma (1. Here as usual for UMS, the parameter gamma depends linearly on the temperature. The second moment of Rn, the variance vn, displays different behaviour corresponding to strongly transient ( gamma >2), not strongly transient (1( gamma (2) and recurrent ( gamma (1) random walks. All results are verified by comparison to computer data obtained through simulations and through numerical series expansions.
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